Settings for the Discretization Sections

There are two categories of discretization — a section on the physics interface node’s window (described here) and adding a Discretization (Node) for global equation-based modeling. To enable the other settings beyond the element order and shape function type and to be able to add separate nodes, click the button (

) and select in the dialog box.

The settings described here are:

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Element Order and Shape Function Type

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Discretization of Fluids

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Accurate Boundary Fluxes

•

Splitting Complex-Valued Variables

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Element Order and Shape Function Type

The PDE and weak form interfaces have different shape functions available with the associated element order (the order of the shape functions). The element order (or, more precisely, the order of the shape function) directly affects the number of degrees of freedom in the solution and the accuracy of the solution. Increasing the order of the elements roughly corresponds to a uniform mesh refinement. Most physics interfaces use Lagrange elements, which can be of order 1 to 5 (or 1 to 7 for the PDE and weak form interfaces), with 2 being the default order in most cases. Where serendipity elements are available (in the mathematics interfaces and the Solid Mechanics interface, for example, for element orders 2, 3, and, in some cases, 4), they can be more efficient than Lagrange elements of the same order (in terms of number of elements and the solution time) for some mesh element types (especially hexahedral meshes), but they can also be more sensitive to distorted mesh elements.

The software adapts the order of the numerical integration to the element orders for the physics in the model. Some physics interfaces use special element types or a reduced element order for some of the field variables. Select the and the as, in most cases, , , , , or (for order 1–5, respectively).

Table 16-2 is an overview of the available shape function types and the element orders supported.


	Not all shape functions are available for all space dimensions and types of equations, and not all shape functions support all orders.

Table 16-2: Shape Function Types
Name	Order	Comments
Lagrange	1–5 or 1–7. Default: 2	The default type
Hermite	3–7. Default: 3
Argyris	Order 5 only.	2D only
Discontinuous Lagrange	0–7. Default: 2
Nodal discontinuous Lagrange	1–10 (1D and 2D); 1–7 (3D). Default: 2	Special shape functions for wave equations
Discontinuous scalar density	0–7. Default: 2	Not available on boundaries, edges, or points
Bubble	2 (1D); 3 (2D); 4 (3D)	Lower order on boundaries, edges, and points
Gauss point data	0, 2, 4, 6, 8, 10, 12, or 14. Default: 4	Discrete values associated with the quadrature points in an integration rule of the given order
Nodal serendipity	2–4	In the Solid Mechanics interface, order 2 and 3 are available as Quadratic serendipity and Cubic serendipity, respectively.
Divergence	1–4 (1D and 2D); 1–3 (3D). Default: 2	For vector fields only (1, 2, and 3 dependent variables in 1D, 2D, and 3D, respectively)
Curl	1–4 (2D); 1–3 (3D). Default: 2	2D and 3D only. For vector fields only (1, 2, and 3 dependent variables in 1D, 2D, and 3D, respectively)


	Additional information is included in Elements and Shape Functions chapter in this manual and in Elements and Shape Function Programming in the COMSOL Multiphysics Programming Reference Manual.

Discretization of Fluids

The following is an example of the choices of element order for Fluid Flow interfaces:

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means linear elements for both the velocity components and the pressure field. Linear elements are computationally cheaper than higher-order elements and are also less prone to introducing spurious oscillations, thereby improving the numerical robustness. In other words, this can be computationally efficient but requires streamline stabilization of the Navier-Stokes equations. This is the default element order for the Laminar Flow and Turbulent Flow single-phase flow interfaces and the discretization of fluids in the multiphase flow interfaces.

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means second-order elements for the velocity components and linear elements for the pressure field. Second-order elements work well for low flow velocities. This is the default element order for the Creeping Flow interface.

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means third-order elements for the velocity components and second-order elements for the pressure field. This can add additional accuracy but it also adds additional degrees of freedom compared to P2+P1 elements.


	The abbreviation PmPn is often used to indicate the polynomial order of, in this case, the shape functions (elements) for the velocity components (m) and the pressure (n) when using tetrahedral or triangular elements. Here a corresponding nomenclature is used for all element shapes. The theory about this is in P.M. Gresho and R.L. Sani, Incompressible Flow and the Finite Element Method, Volume 2: Isothermal Laminar Flow, John Wiley & Sons, 2000.


	The discretization of the temperature field follows that of the fluid’s velocity components, so the temperature order is 1 for P1+P1, 2 for P2+P1, and 3 for P3+P2.

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Numerical Stability — Stabilization Techniques for Fluid Flow

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Numerical Stabilization

Accurate Boundary Fluxes

Some physics can create and compute variables that accurately represent the flux across all boundaries. To enable these variables, select the check box. Optionally, the smoothing can provide a more well-behaved flux value close to singularities. You add smoothing by selecting the check box.


	Computing Accurate Fluxes

Splitting Complex-Valued Variables

From the list, you can specify the value type ( or of dependent variables when the setting is activated in the node of any solver sequence used. The default is the complex value type, but you can specify that the value of a dependent variable is real to make sure that it does not get affected by small imaginary contributions, which can occur, for example, when combining a Time Dependent or Stationary study with a frequency-domain study. If the split complex variables setting is not active, the value type is ignored.


	For information about how to specify the splitting of complex variables, see Compile Equations.


	By default, the general number formatting algorithm for complex-valued numbers in COMSOL Multiphysics ignores small real or imaginary parts. If you want to see also such small real or imaginary parts, open the Preferences dialog box, and under Precision on the General page, clear the Suppress small real or imaginary part check box.